Parallelograms

Question

In parallelogram ABCD, AC and BD intersect at E. If AE is 5x-6 and EC is 2x+3, find the of x. What is the length of AC? If BD is 6x, what kind of quadrilateral could his be? Why?

Answer 1

The diagonals of a parallelogram bisect each other.  The point E is therefore the midpoint of AC.  As the midpoint, AE ≅ EC, so

 

mAE = mEC

5x - 6 = 2x +3     Isolate the variable

5x - 2x = 3 + 6

3x = 9

x = 3

 

The length, mAC  = mAE + mEC = (5x - 6) + (2x + 3) = 7x - 3 

since x = 3

mAC = 7(3) - 3 = 21 - 3 = 18

 

If the other diagonal, BD, has the measure mBD = 6x, and x = 3, then

mBD = 6(3) = 18

 

Since both diagonals have the same length, mAC = 18 = mBD, the Parallelogram ABCD is a rectangle.  

 

(It is not specified, but if AC intersects BD at right angles then the parallelogram ABCD would be a square.)